Anomalous diffusion and Lévy random walk of magnetic field lines in three dimensional turbulence

Abstract

The transport of magnetic field lines is studied numerically where three dimensional (3-D) magnetic fluctuations, with a power law spectrum, and periodic over the simulation box are superimposed on an average uniform magnetic field. The weak and the strong turbulence regime, δB∼B0, are investigated. In the weak turbulence case, magnetic flux tubes are separated from each other by percolating layers in which field lines undergo a chaotic motion. In this regime the field lines may exhibit Lévy, rather than Gaussian, random walk, changing from Lévy flights to trapped motion. The anomalous diffusion laws 〈Δx2i〉∝sα with α≳1 and α<1, are obtained for a number of cases, and the non-Gaussian character of the field line random walk is pointed out by computing the kurtosis. Increasing the fluctuation level, and, therefore stochasticity, normal diffusion (α≂1) is recovered and the kurtoses reach their Gaussian value. However, the numerical results show that neither the quasi-linear theory nor the two dimensional percolation theory can be safely extrapolated to the considered 3-D strong turbulence regime.